microplanes

Microplanes are a class of constitutive models used in solid mechanics to describe the behavior of materials by averaging responses over a large set of imaginary planes, or microplanes, within a material element. Each microplane has a distinct orientation, and material response is defined on the plane rather than only at a single point in the bulk.

In a microplane formulation, the macroscopic strain tensor is projected onto each microplane to obtain a normal and tangential strain on that plane. A constitutive law, which may be nonlinear and include damage or plasticity, is then defined on the microplane to determine the corresponding stresses. The overall, or macro, stress of the material is obtained by integrating or averaging the microplane stresses over all orientations, often with an assumed isotropic distribution of microplanes. This framework allows the macroscopic response to emerge from combined normal and shear behaviors on many orientations.

Microplane models are particularly used for quasi-brittle materials such as concrete, rock, and ceramics, where damage, cracking, and softening are important. They can capture phenomena such as strain softening, size effects, and rate dependence more naturally than some traditional models. Variants and specific implementations, including versions known as M4 and related formulations, differ in the choice of microplane laws, damage criteria, and how tangential interactions are treated.

Numerically, microplane models are implemented within finite element or other continuum mechanics codes by performing numerical integration over the unit sphere to account for all microplane orientations. Calibration relies on experimental data from different loading paths to determine parameters governing normal damage, shear resistance, and rate effects.

See also: continuum damage mechanics, constitutive modeling, quasi-brittle materials.